Gauge-invariant charges of the dual graviton

Chris Hull; Maxwell L. Vel\'asquez Cotini Hutt; Ulf Lindstr\"om

REVIEW 3 major objections 3 minor 28 references

Gauge-invariant charges of the dual graviton

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The dual graviton's charges equal the graviton's on-shell charges.

desk verdict A careful extension of the authors' charge programme to the dual graviton, with a real but localized coefficient slip in the d=4 interpretation that does not break the central computation. read the letter →

arxiv 2412.10503 v2 pith:3JWGBP62 submitted 2024-12-13 hep-th gr-qc

Chris Hull , Ulf Lindström , Maxwell L. Velásquez Cotini Hutt This is my paper · ORCID

classification hep-thgr-qc

keywords dualgravitonlinearisedgravityPenrosechargesconformalKilling-Yanotensorsgeneralizedsymmetriesmagnetic1-formgravitationalduality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Linearised Einstein gravity on Minkowski space admits an equivalent description in terms of a dual graviton field, a [d-3,1] tensor gauge field. This paper proves that the dual formulation has its own set of conserved charges, the dual Penrose charges, that remain well defined even when the spacetime has punctures or compact directions, where the naive Noether charges $Q[\lambda]$ and $Q[\kappa]$ are not gauge-invariant. The central result is the identity $\tilde Q[K] = Q[\lambda] + \int_\Sigma d\star\tilde Z[K]$, together with the on-shell equality $Q[K] = \tilde Q[K]$ of the Penrose charges of the graviton and dual graviton formulations. A sympathetic reader should care because it shows that the genuine 1-form and $(d-3)$-form symmetries of linearised gravity are the same in both formulations, and because the electric/magnetic classification of the charges differs from p-form gauge theory: some charges are magnetic in both descriptions, and some are electric in both.

What carries the argument

The load-bearing object is the conformal Killing-Yano (CKY) 2-form $K$, whose general solution on Minkowski space splits into constant (A), translational (B), linear (C), and quadratic (D) pieces. $K$ determines the improved graviton current $Y_+[K]$ and, through the duality identification $\tilde S(D)=R(h)$, the dual improved Penrose 2-form $\tilde Y_+[K] = \tilde S^{\mu\nu|\rho\sigma}K_{\rho\sigma}$. The paper's central identity is (3.21)-(3.23): $\tilde Y_+[K] = J[\lambda] + d\tilde Z[K]$, and after integration $\tilde Q[K] = Q[\lambda] + \int_\Sigma d\star\tilde Z[K]$; on-shell the Penrose and dual Penrose charges coincide, $Q[K] = \tilde Q[K]$. The machinery also includes the generalised Killing tensors $\lambda$ and $\kappa$ of the dual graviton, the 3-forms $Z[K]$ and $\tilde Z[K]$ appearing as improvement terms, and the tensor $V$ built from $\kappa$ that carries the covariant current $\Omega[V]$.

What would settle it

Compute both sides of (3.23) for an explicit non-globally-defined dual graviton, e.g. the four-dimensional configuration $D_{it}=2A_i$ with $F=dA=-\frac{M}{2}\operatorname{Vol}(S^2)$, the dual of linearised Schwarzschild. The paper predicts $\tilde Q[K]=Q[B]=-2\pi M B_0$ and also $\tilde Q[K]=Q[\lambda]+\int_{S^2} d\star\tilde Z[K]$ with the improvement term $B_0\int F$. Evaluating the left side directly from (3.17) and the right side from (2.42) and (3.22) and finding different numbers would falsify the central identity; agreement would confirm it.

Watch

Extended reading notes

Core claim

The paper claims that for the free graviton theory there is a complete, gauge-invariant charge dictionary between the graviton formulation and the dual graviton formulation. In the graviton picture, the improved Penrose 2-form $Y_+[K]$ built from the linearised Riemann tensor and a conformal Killing-Yano 2-form $K$ gives gauge-invariant charges $Q[K]$. In the dual picture, the analogous current $\tilde Y_+[K] = \tilde S^{\mu\nu|\rho\sigma}K_{\rho\sigma}$ built from the dual field strength gives dual Penrose charges $\tilde Q[K]$, and these are equal on-shell to $Q[K]$. The paper proves the key identity $\tilde Q[K] = Q[\lambda] + \int_\Sigma d\star\tilde Z[K]$, so the gauge-invariant dual charge is the Noether charge for the dual graviton's $\lambda$-invariance plus a topological term that vanishes when the dual graviton is globally defined but repairs gauge dependence when it is not. It also constructs $\tilde\Omega[V] = J[\kappa] + d\tilde\Phi$, giving a gauge-invariant completion $\tilde Q[V]$ of the $\kappa$-charges. Consequently the gauge-invariant charges generate the same 1-form and $(d-3)$-form symmetries in both descriptions, with A-type CKY tensors magnetic in both formulations and D-type electric in both.

Load-bearing premise

The charge dictionary assumes the duality identification $\tilde S(D)=R(h)$ and the companion gauge choice (2.19) hold even when the fields are only locally defined on punctured or compact Minkowski space; this equivalence is imported from earlier work rather than derived here, and if it fails the equality $Q[K]=\tilde Q[K]$ and the electric/magnetic classification would not follow.

Editorial extensions

If this is right

The gauge-invariant charges $Q[K]$ and $\tilde Q[K]$ generate the same 1-form symmetries in the graviton and dual graviton formulations, so the two descriptions have identical symmetry content even on spacetimes with punctures or compact directions.
In $d=4$ there are 20 independent gauge-invariant charges; in $d>4$ the A- and C-type Penrose charges vanish on-shell, leaving the B- and D-type charges that generate a group $\mathbb{R}^{d(d+1)/2}$ of 1-form symmetries and an equal number of $(d-3)$-form symmetries.
The A-type charges are magnetic in both formulations, so the objects carrying them are intrinsically solitonic; the D-type charges are electric in both, so their carriers are intrinsically electric, behaviour impossible in standard p-form gauge theory.
The $Q[\lambda]$ charges of the dual graviton equal the gauge-invariant dual Penrose charge plus a topological term; in $d>4$ the constant-$\lambda$ charges become topological and vanish when $D$ is globally defined, whereas in $d=4$ they remain genuine electric charges.
Linearised Schwarzschild and Taub-NUT give a concrete duality pair: the B-type charge is the ADM mass in the $h$ description and a magnetic charge $\int F = -2\pi M B_0$ in the $D$ description.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If correct, the improvement terms identify which linearised ADM charges remain meaningful when the graviton is only locally defined; the same cohomological criterion could be applied to nonlinear asymptotically flat spacetimes to decide when ADM charges should be replaced by Penrose-type integrals.
The equality $Q[K]=\tilde Q[K]$ suggests the duality acts trivially on the charge lattice of linearised gravity but nontrivially on which charges look electric or magnetic; this pattern may persist for higher-spin dual gauge fields, where electric/magnetic roles would again be fixed per CKY type.
The paper leaves the mixed 't Hooft anomalies of the dual 1-form symmetries to future work; a direct check that the anomaly polynomial computed from $\tilde Y_+[K]$ matches that from $Y_+[K]$ would be a sharp test of the whole picture.
Because the free theory has no local interacting dual graviton, the charges constructed here are strictly linearised objects; extending them to full Einstein gravity would require defining the magnetic charges as non-local or hidden symmetries.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper does something real: it constructs gauge-invariant conserved currents for the dual graviton, shows they differ from the Noether charges Q[λ] by a topological term (eq 3.23), and works out the electric/magnetic classification under duality. The dual improved Penrose currents and the charge relations (3.23), (3.56) are new relative to the graviton-only results in the authors' earlier papers. The appendix gives a full algebraic derivation of the central identity (3.21), and the Schwarzschild/Taub-NUT pair in section 4.3 explicitly checks the B-type charge relation. That is serious, reproducible evidence.

The main soft spot is a genuine coefficient inconsistency in section 4.1. For B-type CKY tensors, (3.6) and (3.9) give k = -(d-3)B, i.e. -B in d=4, but section 4.1 says k=-B/2. Similarly, for D-type, (3.9) gives k=2D·x in d=4, while section 4.1 says k=D·x. These are exactly the relations used in the classification narrative. The factor errors do not appear to infect the central computation: the appendix derivation of (3.21)-(3.23) uses the definitions directly, and the worked example in 4.3 matches the correct (3.9) normalization, not the wrong 4.1 values. So the stress-test worry that a factor error in \tilde Z[K] could break Q[K]=tilde Q[K] does not land on the actual paper; the inconsistency is localized to the interpretation section, but it is still a real flaw that must be fixed.

Two smaller caveats. The duality identification (2.18) is imported from prior work, not re-derived; that is acceptable for a paper in a continuing program, but it means the central equality (4.3) inherits whatever topology assumptions that identification carries. And the jump from conserved surface charges to 'topological operators generating 1-form symmetries' is asserted rather than constructed; readers who want the operator story should look elsewhere.

Overall: a solid, careful paper that extends the authors' programme and gives a new classification. The interpretation section needs a coefficient correction, and the duality assumptions should be flagged more explicitly. For a journal like JHEP or CQG, this deserves peer review; a referee should check the section 4.1 factors and the appendix normalization, but the core seems sound. I would bring it to our reading group.

Referee Report

3 major / 3 minor

Summary. The paper constructs gauge-invariant conserved 2-form currents and charges in the dual graviton formulation of linearised gravity. It defines improved dual Penrose currents \tilde Y_+[K] and \tilde\Omega[V], proves the algebraic relation (3.21) between \tilde Y_+[K] and the secondary Noether current J[\lambda], and derives the corresponding charge relation (3.23). It then analyses the four types of CKY 2-forms, argues on-shell equality Q[K]=\tilde Q[K] via the duality (2.18), and interprets the charges as electric or magnetic in the two formulations, including the claim that A-type charges are magnetic in both formulations and D-type charges are electric in both. A linearised Schwarzschild/Taub-NUT pair is used to illustrate the B-type duality.

Significance. If the central identities are correct, the paper gives a useful and systematic map between gauge-invariant charges in the graviton and dual-graviton formulations, with direct consequences for the counting of 1-form and (d-3)-form symmetries. The explicit relation (3.21), the summary tables, and the Schwarzschild/Taub-NUT example are valuable, and the paper is careful to distinguish identically conserved currents from on-shell conserved ones. The main weaknesses are coefficient errors in the interpretation section and the absence of an independent check of the long appendix derivation, so the quantitative content of the charge relations needs revision before the claims can be fully trusted.

major comments (3)

[Section 4.1, Eqs. (4.5) and surrounding text] The Killing vectors quoted for B- and D-type CKY tensors are inconsistent with Eqs. (3.5), (3.6), and (3.9). For K_{\mu\nu}=B_{[\mu}x_{\nu]}, Eq. (3.6) gives \hat K_\mu=-(1/2)B_\mu, so Eq. (3.9) gives k_\mu=2(d-3)\hat K_\mu=-B_\mu in d=4, not -B/2 as stated. For D-type tensors, (3.9) gives k_\mu=2D_{\mu\nu}x^\nu, not D_{\mu\nu}x^\nu as stated. These factors enter the identification of B-type Penrose charges with ADM momenta and the equality (4.5), so the interpretation in Section 4.1 and in Table 1 is quantitatively unreliable unless the normalisation of k in (3.9) is changed consistently throughout the paper.
[Appendix A and Eq. (3.21)] The central relation (3.21), with \tilde Z[K] given in (3.22), is the basis for (3.23) and for the subsequent electric/magnetic classification, but the derivation in Appendix A is long and not independently checked. At least one displayed equation, (A.10), contains an index repetition in the epsilon symbol, and the passage from (A.14) to (3.21) is not shown in enough detail to verify the relative signs and coefficients. The Schwarzschild/Taub-NUT example in Section 4.3 is not sufficient to check the pointwise identity: Eq. (4.15) asserts \int d\star\tilde Z[K]=B_0\int F without displaying the intermediate factors. Please provide an independent check of (3.21), for example by substituting (3.21) and verifying term-by-term that its divergence reproduces (3.20), or by an explicit algebraic computation of the coefficients in (3.22).
[Sections 2.2, 4, and 4.3] The equality Q[K]=\tilde Q[K] in (4.3) and the topological-charge identifications in (3.23) and (4.5) are derived under the duality identification (2.18) and the gauge choice (2.19). The paper does not state explicitly how these identifications behave when h or D is only locally defined, which is precisely the situation in which the total-derivative terms are non-trivial. Since (2.18) is taken from previous work rather than re-derived here, the central claims inherit this assumption. Please state the domain of validity of (2.18)-(2.19) for non-globally-defined field configurations and specify whether the transition functions of h and D are assumed to be related by the duality gauge.

minor comments (3)

[Section 5.2, Eq. (5.1)] In Eq. (5.1), the charge in the dual graviton formulation should be denoted \tilde Q[V], not Q[V]; Section 3.6 carefully distinguishes Q[V] in the graviton formulation from \tilde Q[V] in the dual formulation, and reusing Q[V] in (5.1) is confusing.
[Appendix A, Eq. (A.10)] The displayed equation after (A.10) has an index typo in the epsilon symbol, and the symbol 'dim' is used in (A.7) without being defined; it should be d throughout for consistency.
[Section 3.5, Eq. (3.46)] The sentence after (3.46) contains a grammatical slip ('However, the since the Hodge dual...'), and the displayed identity would be easier to follow with a brief explanation of the epsilon contraction used in the final equality.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the charge relations are derived by explicit algebra from defined currents, with prior self-citations supplying background lemmas rather than the target result.

full rationale

The central new result is Eq. (3.23), tilde Q[K] = Q[lambda] + integral over Sigma of d * tilde Z[K]. It is obtained by integrating Eq. (3.21), which is proven in Appendix A by substituting the definitions of tilde S (2.8), the CKY equation (3.2), the secondary Noether current J[lambda] (2.42), and the identity (A.12) from [5]. None of these inputs assumes Eq. (3.21) or the charge equality; the derivation is a direct computation. The on-shell equality (4.3), Q[K] = tilde Q[K], follows from the duality identification (2.18) together with the on-shell simplifications (4.1)-(4.2), so it is a consequence of the duality map rather than a renamed version of it. The duality identification itself is not re-derived in this paper, but it is argued locally from matching Bianchi identities and field equations, and the cited prior work [1,2,3] states assumptions that do not include the target charge relations. This is legitimate reliance on prior results, not circularity. The same applies to the use of covariant currents from [5,6]. No parameter is fitted to a subset of data and then called a prediction; no quantity is defined in terms of the result it is used to establish. The factor discrepancies noted in Section 4.1 relative to Eq. (3.9) are potential correctness issues in the interpretation of the charges, but they do not make the derivation circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or new physical entities. The derivation rests on the dual-graviton formalism and CKY tensor technology taken from earlier work, the global duality identification between h and D, and the assumption of topologically non-trivial backgrounds. All charges are defined via closed forms, so there is no numerical fitting.

assumptions (5)

domain assumption The duality identification tilde S(D) = R(h) can be imposed globally, with gauge choice Gamma(h) = -(1/2) star tilde Gamma(D) (eqs (2.18), (2.19)).
Used throughout to map charges between formulations, including eq (4.3). It is established in prior work [1,2,3] and not re-derived here.
domain assumption Minkowski space with points or regions removed is the background, and gauge fields are defined only locally with transition functions supporting non-trivial topology.
The distinction between global and local field configurations is what makes the topological and magnetic charges non-vanishing; stated in section 1 and used in every charge integral.
standard math The general solution to the conformal Killing-Yano equation on Minkowski space is (3.5), and every Killing-Yano 2-form is the divergence of a CKY 3-form.
Used in section 3.3 to prove that A-type and C-type dual Penrose charges vanish for d > 4; cited to [5,8,19,24].
domain assumption Higher-form symmetries come in dual pairs with equal numbers of 1-form and (d-3)-form symmetries.
Assumed from [25] in section 3.5 to check consistency of the charge counting; it is a background principle of the generalized-symmetries framework, not proven here.
domain assumption The field equations are imposed throughout, with sources U = 0 and T = 0 in regions where charges are defined.
Conservation of the currents and on-shell equalities such as (2.14) rely on this; stated in section 2.1.

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Cite this review

Pith. "Pith review of Gauge-invariant charges of the dual graviton." pith.science (2026). https://pith.science/paper/3JWGBP62

@misc{pith2026241210503,
  author       = {Pith},
  title        = {Pith review of: Gauge-invariant charges of the dual graviton},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JWGBP62}},
  note         = {Machine review of arXiv:2412.10503}
}

read the original abstract

The free graviton theory given by linearising Einstein's theory has a dual formulation in terms of a dual graviton field. The dual graviton theory has two gauge invariances giving rise to two conserved charges, while the ADM charges of the graviton theory become magnetic charges for the dual graviton theory. These charges can be ill-defined in topologically non-trivial settings and we find improvement terms that can be added to these to give gauge-invariant conserved charges. These gauge-invariant charges, which have local expressions in both the graviton and dual graviton formulation, give topological operators of the theory that should be considered as the generators of the genuine symmetries of the theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Hull, Magnetic charges for the graviton , JHEP 05 (2024) 257 [ 2310.18441]

C.M. Hull, Magnetic charges for the graviton , JHEP 05 (2024) 257 [ 2310.18441]

arXiv 2024

[2] [2]

Hull, Strongly coupled gravity and duality , Nucl

C.M. Hull, Strongly coupled gravity and duality , Nucl. Phys. B 583 (2000) 237 [hep-th/0004195]

arXiv 2000

[3] [3]

Hull, Duality in gravity and higher spin gauge fields , JHEP 09 (2001) 027 [hep-th/0107149]

C.M. Hull, Duality in gravity and higher spin gauge fields , JHEP 09 (2001) 027 [hep-th/0107149]

arXiv 2001

[4] [4]

Hull, Covariant action for self-dual p-form gauge fields in general spacetimes , JHEP 04 (2024) 011 [ 2307.04748]

C.M. Hull, Covariant action for self-dual p-form gauge fields in general spacetimes , JHEP 04 (2024) 011 [ 2307.04748]

arXiv 2024

[5] [5]

Hull, M.L

C. Hull, M.L. Hutt and U. Lindstr¨ om, Charges and topology in linearised gravity , JHEP 07 (2024) 097 [ 2401.17361]

arXiv 2024

[6] [6]

Hull, M.L

C. Hull, M.L. Hutt and U. Lindstr¨ om, Gauge-invariant magnetic charges in linearised gravity, Class. Quant. Grav. 41 (2024) 195012 [ 2405.08876]

arXiv 2024

[7] [7]

Penrose, Quasilocal mass and angular momentum in general relativity , Proc

R. Penrose, Quasilocal mass and angular momentum in general relativity , Proc. Roy. Soc. Lond. A 381 (1982) 53. – 29 –

work page 1982

[8] [8]

Hinterbichler, D.M

K. Hinterbichler, D.M. Hofman, A. Joyce and G. Mathys, Gravity as a gapless phase and biform symmetries , JHEP 02 (2023) 151 [ 2205.12272]

arXiv 2023